Polytope of Type {90,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,4}*720c
if this polytope has a name.
Group : SmallGroup(720,397)
Rank : 3
Schlafli Type : {90,4}
Number of vertices, edges, etc : 90, 180, 4
Order of s0s1s2 : 45
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{90,4,2} of size 1440
Vertex Figure Of :
{2,90,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {45,4}*360
3-fold quotients : {30,4}*240c
5-fold quotients : {18,4}*144c
6-fold quotients : {15,4}*120
10-fold quotients : {9,4}*72
15-fold quotients : {6,4}*48b
30-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {90,4}*1440
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)
( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)
( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)
( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)
( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)
( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)
( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)
( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)
(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)
(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)
(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)(194,231)
(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)(202,235)
(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)(210,227)
(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)(242,307)
(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)(250,311)
(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)(258,351)
(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)(266,343)
(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)(274,347)
(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)(282,327)
(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)(290,319)
(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)(298,323)
(299,322)(300,324);;
s1 := ( 1,253)( 2,254)( 3,256)( 4,255)( 5,261)( 6,262)( 7,264)( 8,263)
( 9,257)( 10,258)( 11,260)( 12,259)( 13,241)( 14,242)( 15,244)( 16,243)
( 17,249)( 18,250)( 19,252)( 20,251)( 21,245)( 22,246)( 23,248)( 24,247)
( 25,289)( 26,290)( 27,292)( 28,291)( 29,297)( 30,298)( 31,300)( 32,299)
( 33,293)( 34,294)( 35,296)( 36,295)( 37,277)( 38,278)( 39,280)( 40,279)
( 41,285)( 42,286)( 43,288)( 44,287)( 45,281)( 46,282)( 47,284)( 48,283)
( 49,265)( 50,266)( 51,268)( 52,267)( 53,273)( 54,274)( 55,276)( 56,275)
( 57,269)( 58,270)( 59,272)( 60,271)( 61,193)( 62,194)( 63,196)( 64,195)
( 65,201)( 66,202)( 67,204)( 68,203)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,181)( 74,182)( 75,184)( 76,183)( 77,189)( 78,190)( 79,192)( 80,191)
( 81,185)( 82,186)( 83,188)( 84,187)( 85,229)( 86,230)( 87,232)( 88,231)
( 89,237)( 90,238)( 91,240)( 92,239)( 93,233)( 94,234)( 95,236)( 96,235)
( 97,217)( 98,218)( 99,220)(100,219)(101,225)(102,226)(103,228)(104,227)
(105,221)(106,222)(107,224)(108,223)(109,205)(110,206)(111,208)(112,207)
(113,213)(114,214)(115,216)(116,215)(117,209)(118,210)(119,212)(120,211)
(121,317)(122,318)(123,320)(124,319)(125,313)(126,314)(127,316)(128,315)
(129,321)(130,322)(131,324)(132,323)(133,305)(134,306)(135,308)(136,307)
(137,301)(138,302)(139,304)(140,303)(141,309)(142,310)(143,312)(144,311)
(145,353)(146,354)(147,356)(148,355)(149,349)(150,350)(151,352)(152,351)
(153,357)(154,358)(155,360)(156,359)(157,341)(158,342)(159,344)(160,343)
(161,337)(162,338)(163,340)(164,339)(165,345)(166,346)(167,348)(168,347)
(169,329)(170,330)(171,332)(172,331)(173,325)(174,326)(175,328)(176,327)
(177,333)(178,334)(179,336)(180,335);;
s2 := ( 1,184)( 2,183)( 3,182)( 4,181)( 5,188)( 6,187)( 7,186)( 8,185)
( 9,192)( 10,191)( 11,190)( 12,189)( 13,196)( 14,195)( 15,194)( 16,193)
( 17,200)( 18,199)( 19,198)( 20,197)( 21,204)( 22,203)( 23,202)( 24,201)
( 25,208)( 26,207)( 27,206)( 28,205)( 29,212)( 30,211)( 31,210)( 32,209)
( 33,216)( 34,215)( 35,214)( 36,213)( 37,220)( 38,219)( 39,218)( 40,217)
( 41,224)( 42,223)( 43,222)( 44,221)( 45,228)( 46,227)( 47,226)( 48,225)
( 49,232)( 50,231)( 51,230)( 52,229)( 53,236)( 54,235)( 55,234)( 56,233)
( 57,240)( 58,239)( 59,238)( 60,237)( 61,244)( 62,243)( 63,242)( 64,241)
( 65,248)( 66,247)( 67,246)( 68,245)( 69,252)( 70,251)( 71,250)( 72,249)
( 73,256)( 74,255)( 75,254)( 76,253)( 77,260)( 78,259)( 79,258)( 80,257)
( 81,264)( 82,263)( 83,262)( 84,261)( 85,268)( 86,267)( 87,266)( 88,265)
( 89,272)( 90,271)( 91,270)( 92,269)( 93,276)( 94,275)( 95,274)( 96,273)
( 97,280)( 98,279)( 99,278)(100,277)(101,284)(102,283)(103,282)(104,281)
(105,288)(106,287)(107,286)(108,285)(109,292)(110,291)(111,290)(112,289)
(113,296)(114,295)(115,294)(116,293)(117,300)(118,299)(119,298)(120,297)
(121,304)(122,303)(123,302)(124,301)(125,308)(126,307)(127,306)(128,305)
(129,312)(130,311)(131,310)(132,309)(133,316)(134,315)(135,314)(136,313)
(137,320)(138,319)(139,318)(140,317)(141,324)(142,323)(143,322)(144,321)
(145,328)(146,327)(147,326)(148,325)(149,332)(150,331)(151,330)(152,329)
(153,336)(154,335)(155,334)(156,333)(157,340)(158,339)(159,338)(160,337)
(161,344)(162,343)(163,342)(164,341)(165,348)(166,347)(167,346)(168,345)
(169,352)(170,351)(171,350)(172,349)(173,356)(174,355)(175,354)(176,353)
(177,360)(178,359)(179,358)(180,357);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(360)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 49)( 14, 51)
( 15, 50)( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)
( 23, 54)( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)
( 31, 46)( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)
( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)
( 71,130)( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)
( 79,170)( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)
( 87,162)( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)
( 95,166)( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)
(103,146)(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)
(111,138)(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)
(119,142)(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)
(194,231)(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)
(202,235)(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)
(210,227)(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)
(242,307)(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)
(250,311)(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)
(258,351)(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)
(266,343)(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)
(274,347)(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)
(282,327)(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)
(290,319)(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)
(298,323)(299,322)(300,324);
s1 := Sym(360)!( 1,253)( 2,254)( 3,256)( 4,255)( 5,261)( 6,262)( 7,264)
( 8,263)( 9,257)( 10,258)( 11,260)( 12,259)( 13,241)( 14,242)( 15,244)
( 16,243)( 17,249)( 18,250)( 19,252)( 20,251)( 21,245)( 22,246)( 23,248)
( 24,247)( 25,289)( 26,290)( 27,292)( 28,291)( 29,297)( 30,298)( 31,300)
( 32,299)( 33,293)( 34,294)( 35,296)( 36,295)( 37,277)( 38,278)( 39,280)
( 40,279)( 41,285)( 42,286)( 43,288)( 44,287)( 45,281)( 46,282)( 47,284)
( 48,283)( 49,265)( 50,266)( 51,268)( 52,267)( 53,273)( 54,274)( 55,276)
( 56,275)( 57,269)( 58,270)( 59,272)( 60,271)( 61,193)( 62,194)( 63,196)
( 64,195)( 65,201)( 66,202)( 67,204)( 68,203)( 69,197)( 70,198)( 71,200)
( 72,199)( 73,181)( 74,182)( 75,184)( 76,183)( 77,189)( 78,190)( 79,192)
( 80,191)( 81,185)( 82,186)( 83,188)( 84,187)( 85,229)( 86,230)( 87,232)
( 88,231)( 89,237)( 90,238)( 91,240)( 92,239)( 93,233)( 94,234)( 95,236)
( 96,235)( 97,217)( 98,218)( 99,220)(100,219)(101,225)(102,226)(103,228)
(104,227)(105,221)(106,222)(107,224)(108,223)(109,205)(110,206)(111,208)
(112,207)(113,213)(114,214)(115,216)(116,215)(117,209)(118,210)(119,212)
(120,211)(121,317)(122,318)(123,320)(124,319)(125,313)(126,314)(127,316)
(128,315)(129,321)(130,322)(131,324)(132,323)(133,305)(134,306)(135,308)
(136,307)(137,301)(138,302)(139,304)(140,303)(141,309)(142,310)(143,312)
(144,311)(145,353)(146,354)(147,356)(148,355)(149,349)(150,350)(151,352)
(152,351)(153,357)(154,358)(155,360)(156,359)(157,341)(158,342)(159,344)
(160,343)(161,337)(162,338)(163,340)(164,339)(165,345)(166,346)(167,348)
(168,347)(169,329)(170,330)(171,332)(172,331)(173,325)(174,326)(175,328)
(176,327)(177,333)(178,334)(179,336)(180,335);
s2 := Sym(360)!( 1,184)( 2,183)( 3,182)( 4,181)( 5,188)( 6,187)( 7,186)
( 8,185)( 9,192)( 10,191)( 11,190)( 12,189)( 13,196)( 14,195)( 15,194)
( 16,193)( 17,200)( 18,199)( 19,198)( 20,197)( 21,204)( 22,203)( 23,202)
( 24,201)( 25,208)( 26,207)( 27,206)( 28,205)( 29,212)( 30,211)( 31,210)
( 32,209)( 33,216)( 34,215)( 35,214)( 36,213)( 37,220)( 38,219)( 39,218)
( 40,217)( 41,224)( 42,223)( 43,222)( 44,221)( 45,228)( 46,227)( 47,226)
( 48,225)( 49,232)( 50,231)( 51,230)( 52,229)( 53,236)( 54,235)( 55,234)
( 56,233)( 57,240)( 58,239)( 59,238)( 60,237)( 61,244)( 62,243)( 63,242)
( 64,241)( 65,248)( 66,247)( 67,246)( 68,245)( 69,252)( 70,251)( 71,250)
( 72,249)( 73,256)( 74,255)( 75,254)( 76,253)( 77,260)( 78,259)( 79,258)
( 80,257)( 81,264)( 82,263)( 83,262)( 84,261)( 85,268)( 86,267)( 87,266)
( 88,265)( 89,272)( 90,271)( 91,270)( 92,269)( 93,276)( 94,275)( 95,274)
( 96,273)( 97,280)( 98,279)( 99,278)(100,277)(101,284)(102,283)(103,282)
(104,281)(105,288)(106,287)(107,286)(108,285)(109,292)(110,291)(111,290)
(112,289)(113,296)(114,295)(115,294)(116,293)(117,300)(118,299)(119,298)
(120,297)(121,304)(122,303)(123,302)(124,301)(125,308)(126,307)(127,306)
(128,305)(129,312)(130,311)(131,310)(132,309)(133,316)(134,315)(135,314)
(136,313)(137,320)(138,319)(139,318)(140,317)(141,324)(142,323)(143,322)
(144,321)(145,328)(146,327)(147,326)(148,325)(149,332)(150,331)(151,330)
(152,329)(153,336)(154,335)(155,334)(156,333)(157,340)(158,339)(159,338)
(160,337)(161,344)(162,343)(163,342)(164,341)(165,348)(166,347)(167,346)
(168,345)(169,352)(170,351)(171,350)(172,349)(173,356)(174,355)(175,354)
(176,353)(177,360)(178,359)(179,358)(180,357);
poly := sub<Sym(360)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope